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Questions tagged [taylor-series]

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Why is Fisher Scoring easier to compute?

In practice, the observed information matrix (Newton-Raphson) is usually replaced by its expectation, known as Fisher scoring. Link: https://en.wikipedia.org/wiki/Scoring_algorithm#Fisher_scoring ...
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121 views

Taylor Series Expansion of Unconditional Expectation

We know that the best 1st order approximation of an unconditional expectation is the following- $$E(y|x)=(E(y)-\beta E(x))+\beta x$$ where $\beta=\frac{\operatorname{Cov}(y,x)}{\operatorname{Var}(x)}...
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427 views

Is there a nice(r) Taylor expansion of the normal quantile function?

Letting $\Phi$ be the CDF of the standard normal, and $\Phi^{-1}$ be the quantile function of the normal, I am looking for the Taylor series expansion of $$ \Phi^{-1}\left(\Phi\left(x\right) + \...
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Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

X-Posted on math.stackexchange, apologies, though I thought this was equally relevant to both communities. I'm wondering if there exists any higher-order SVD for dimensionality reduction. Note that ...
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81 views

linearization of an estiamtor

Suppose we have two variables $x$ and $y$ defined in some population, with all values of $x$ known. A Poisson sample is drawn, with corresponding inclusion probabilities $\pi_k$ that are proportional ...
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49 views

How can you cluster a set of functions with unknown functional forms?

Say you've $N$ functions $f_N(x)$ defined on a regular grid $x$. You don't know the form of $f(x)$, you've only got several realizations of it. The different functions are related to each other ...
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79 views

Multivariate Taylor series for moments of a random variable

In the expectation propagation for the generative aspect model, Minka uses Taylor series for the parameter estimation of the topics $p(w\mid a)$ eq 31. I am a little confused in the last equation. He ...
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14 views

Taylor Series Power Function in R

I'm trying to recreate a compartmental model from Yu et al (2017). In this model there is a expanded taylor series power function that is used that I am not understanding how to code. where W(t) = 0, ...
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1answer
81 views

Iterated estimation of Taylor series

Say your data generating process is given by the function $y=f(x|\theta)$, where $y$ and $x$ represent variables (data) and $\theta$ represent parameter(s). For convergence reasons (e.g. $f(\cdot)$ is ...
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184 views

Understanding a Taylor expansion for the bias of local polynomial regression

I'm trying to understand the proof of an expression for the asymptotic bias in local polynomial regression of degree $p\ge0$. Specifically, I'm distraught with equation $(3.59)$ on page 102 of this ...
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1answer
76 views

Jaynes Probability theory 4.70 (Different answers with Jaynes when using Taylor power series.)

I have read this derivation. $$L(f)\equiv{lng(f|DX)}=nln(f)+(N-n)ln(1-f)+const \;(4.69) $$ expand L(f) in a power series about $\hat{f}$.The first terms as $$L(f) = L(\hat{f}) - \frac{(f-\hat{f})^2}{2\...
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88 views

Estimate mean and variance of pdf from truncated taylor expansion of logarithm of pdf

In a maximum likelihood fit, one estimates the parameter with the mode of the likelihood $L$, and the variance of this estimator with the second derivative of $\log(L)$: $$ \bar\theta = \mathrm{Mode}[...
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607 views

Manually compute linearized standard errors of mean in survey

I am would like to manually compute the Taylor-linearized standard error and 95% confidence interval for the mean of a variable x in a survey with 1-stage primary ...
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191 views

Delta method with mix of continuous and discrete variables

This is my first question on Cross Validated so please bear with me if my question is lagging in any dimension. My question regards how to evaluate a Jacobian matrix when one variable is binary. I ...
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11 views

Derivation of variance of Odds Ratio in a 2x2 table

Assuming that the elements of a 2x2 table (a,b,c,d) are binomial in distribution with a being $Bin(n_1,p_1)$ and c being $Bin(n_2,p_2)$. How can we show that $\hat {var(logOR)} = \frac 1 a + \frac 1 b ...
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33 views

Quadratic Approximation of the binary logistic regression

I am using https://web.stanford.edu/~hastie/Papers/glmnet.pdf package to solve my optimization problem for the Binary Logistic Regression. On page 10 it is stated that the quadratic approximation of ...
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Derive taylor series expansion of df

I was trying to understand ito's lemma. When I came across the taylor series expansion of df(x). df(x) = f'(x) dx + (1/2!) f''(x) (dx)^2 + ... I searched everywhere for the derivation of this but ...
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1answer
30 views

How to use derivatives of a function to better estimate its variance over the domain?

How to use derivatives of a function to better estimate its variance over the domain? I have a scalar smooth function $f(x)$ and a multivariate random variable $x$ with known distribution (e.g. ...
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21 views

Exponential Kth Moment Derivation

I have essentially a mathematical question, relating to deriving the formula for the kth moment of an exponential. I can't seem to work out how we get from the 2nd line to the 3rd line; i.e. the ...
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106 views

Higher order delta / taylor series approximation relationship to normal distribution?

For a normally distributed variable X, one can call on the delta method to provide an asymptotically normally distributed variable for a non-linear function of it, g(X). This is based on a linear ...
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213 views

mixture of Gaussians vs mixture of quadratic denominators (Cauchy)

It is known that mixture of Gaussians are dense in the set of all distribution functions. A 1-dimensional Gaussian has the following density: $$ \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(\omega-\beta)^...
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412 views

Taylor Expansion of Power of Cumulative Log Normal Distribution Function - Show Lagrange Remainder tends to Zero

QUESTION I am looking to find a simplification of the expression below. I have attempted this using the Taylor series. The question then remains if we can show the Lagrange remainder goes to zero. I ...
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41 views

Express $E(x^{\alpha})$ in terms of $E(e^{-\zeta x})$? to a 1st or second order?

I have a random variable, $X$, and am able to find $\mathbf{E}(e^{-\zeta X})$ for many $\zeta$ (through the Laplace transform solving an ODE as this actually evolves over time) Is there any way I can ...